Nuprl Lemma : lcm-property
∀x,y:ℤ. ∃a,b:ℤ. (CoPrime(a,b) ∧ ((x * b) = lcm(x;y) ∈ ℤ) ∧ ((y * a) = lcm(x;y) ∈ ℤ))
Proof
Definitions occuring in Statement :
lcm: lcm(a;b),
coprime: CoPrime(a,b),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
guard: {T},
implies: P ⇒ Q,
sq_type: SQType(T),
uimplies: b supposing a,
lcm: lcm(a;b),
has-value: (a)↓,
true: True,
squash: ↓T,
gcd: gcd(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_nzero: ℤ-o,
eq_int: (i =z j)
Lemmas referenced :
gcd-property,
decidable__equal_int,
gcd_wf,
coprime_wf,
int_subtype_base,
istype-int,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_not_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
int-value-type,
value-type-has-value,
subtype_base_sq,
istype-universe,
true_wf,
squash_wf,
equal_wf,
int_formula_prop_and_lemma,
intformand_wf,
subtype_rel_self,
iff_weakening_equal,
mul-associates,
mul-commutes,
div-cancel-1,
nequal_wf,
zero-div-rem,
btrue_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
dependent_pairFormation_alt,
natural_numberEquality,
unionElimination,
sqequalRule,
productIsType,
universeIsType,
isectElimination,
equalityIstype,
inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
equalitySymmetry,
because_Cache,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
lambdaEquality_alt,
approximateComputation,
independent_pairFormation,
equalityElimination,
multiplyEquality,
promote_hyp,
independent_functionElimination,
equalityTransitivity,
independent_isectElimination,
intEquality,
cumulativity,
instantiate,
callbyvalueReduce,
Error :memTop,
imageMemberEquality,
universeEquality,
imageElimination,
hyp_replacement,
dependent_set_memberEquality_alt
Latex:
\mforall{}x,y:\mBbbZ{}. \mexists{}a,b:\mBbbZ{}. (CoPrime(a,b) \mwedge{} ((x * b) = lcm(x;y)) \mwedge{} ((y * a) = lcm(x;y)))
Date html generated:
2020_05_19-PM-10_02_19
Last ObjectModification:
2019_12_31-AM-10_56_45
Theory : num_thy_1
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